Simpson's Paradox: A Singularity of Statistical and Inductive Inference

TL;DR

This paper offers a comprehensive analysis of Simpson's paradox, integrating philosophical, causal, and probabilistic perspectives to deepen understanding and address unresolved questions in statistical inference.

Contribution

It provides a schematic analysis, new results, and a unifying view of Simpson's paradox, emphasizing the need for combining causal, statistical, and philosophical approaches.

Findings

New schematic analysis of Simpson's paradox in 2x2 tables

Critical questions raised about causal explanations of SP

Highlighting the importance of integrating logic, probability, and statistics

Abstract

The occurrence of Simpson's paradox (SP) in $2\times 2$ contingency tables has been well studied. The present work comprehensively revisits this problem using a combination of philosophical reflections, causal considerations, and probability theory. The first contribution is to provide a schematic analysis of SP in $2\times 2$ contingency tables and present new results, detailed proofs of previous results and a unifying view of the important examples of SP that have been reported in the literature. The second contribution of the paper suggests a new perspective on the surprise element of SP, raises some critical questions regarding the influential causal analyses of SP and provides a broad perspective on logic, probability, and statistics with SP at its center. The upshot of this research is that we need both causal concepts and statistical tools coupled with philosophical analyses to sort out issues regarding SP.